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21.6: Graphing Points and Lines in Two Dimensions
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21.6.1: Finding Residuals
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In the linear regression part of statistics we are often asked to find the residuals. Given a data point and the regression line, the residual is defined by the vertical difference between the observed value of y and y based on the equation of the regression line.
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21.6.2: Find the Equation of a Line given its Graph
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There are two main ways of representing a line: the first is with its graph, and the second is with its equation. In this section we will practice how to find the equation of the line if we are given the graph of the line. The two key numbers in the equation of a line are the slope and the y-intercept. Thus the main steps in finding the equation of a line are finding the slope and finding the y-intercept. In statistics we are often presented with a scatterplot where we can eyeball the line.
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21.6.3: Find y given x and the Equation of a Line
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A line can be thought of as a function, which means that if a value of x is given, the equation of the line produces exactly one value of y; This is particularly useful in regression analysis where the line is used to make a prediction of one variable given the value of the other variable.
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21.6.4: Graph a Line given its Equation
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Often we are given an equation of a line and we want to visualize it. For this reason, it is important to be able to graph a line given its equation. We will look at lines that are in slope intercept form: y=a + bx where a is the y-intercept of the line and b is the slope of the line. The y-intercept is the value of where the line crosses the y-axis. The slope is the rise over run.
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21.6.5: Interpreting the Slope of a Line
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A common issue when we learn about the equation of a line in an algebra is to state the slope as a number, but have no idea what it represents in the real world. The slope of a line is the rise over the run. If the slope is given by an integer or decimal value we can always put it over the number 1. In this case the line rises by the slope when it runs 1. "Runs 1" means that the x value increases by 1 unit. Therefore the slope represents how much y changes when x changes by 1 unit.
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21.6.6: Interpreting the y-intercept of a Line
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Just like the slope of a line, many algebra classes go over the y-intercept of a line without explaining how to use it in the real world. The y-intercept of a line is the value of \(y\) where the line crosses the y-axis. In other words, it is the value of \(y\) when the value of \(x\) is equal to 0. Sometimes this has true meaning for the model that the line provides, but other times it is meaningless. We will encounter examples of both types in this section.
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21.6.7: Plot an Ordered Pair
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We have already gone into detail about how to plot points on a number line, and that is very useful for single variable presentations. Now we will move to questions that involve comparing two variables. Working with two variables is frequently encountered in statistical studies and we would like to be able to display the results graphically. This is best done by plotting points in the xy-plane.